The first two items follows directly from the ordinary case (see, for instance [1]}).
For the item (3) it is sufficient to observe that \(\mbox{Tr}(x) \equiv 0\)  is a trace-identity of \(UT_2\)  but such a polynomial does not vanish on \(A\) , for any \(A \in \lbrace  D_2^{t_{\alpha , \beta }}, D_2^{t_{\gamma ,\gamma }}, D_2^{t_{\delta ,0}}, C_2^{t_{\epsilon ,1}} \rbrace \) . Finally, since the algebras \(D_2^{t_{\alpha , \beta }}, D_2^{t_{\gamma ,\gamma }}, D_2^{t_{\delta ,0}}, C_2^{t_{\epsilon ,1}}\)  are commutative and \(UT_2\)  is not, we get item (4), and the proof is complete.
